A systems-level analysis of perfect adaptation in yeast
osmoregulation
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Muzzey, Dale et al. “A Systems-Level Analysis of Perfect
Adaptation in Yeast Osmoregulation.” Cell 138.1 (2009):
160–171. Web.
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http://dx.doi.org/10.1016/j.cell.2009.04.047
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Author Manuscript
Cell. Author manuscript; available in PMC 2011 June 7.
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Published in final edited form as:
Cell. 2009 July 10; 138(1): 160–171. doi:10.1016/j.cell.2009.04.047.
A systems-level analysis of perfect adaptation in yeast
osmoregulation
Dale Muzzey1,2,4, Carlos A. Gómez-Uribe1,3,4, Jerome T. Mettetal1, and Alexander van
Oudenaarden1,*
1 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2
Harvard University Graduate Biophysics Program, Harvard Medical School, Boston, MA 02115,
USA
3
Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA
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SUMMARY
Negative feedback can serve many different cellular functions, including noise reduction in
transcriptional networks and the creation of circadian oscillations. However, only one special type
of negative feedback (“integral feedback”) ensures perfect adaptation, where steady-state output is
independent of steady-state input. Here we quantitatively measure single-cell dynamics in the
Saccharomyces cerevisiae hyperosmotic shock network, which regulates membrane turgor
pressure. Importantly, we find that the nuclear enrichment of the MAP kinase Hog1 perfectly
adapts to changes in external osmolarity, a feature robust to signaling fidelity and operating with
very low noise. By monitoring multiple system quantities (e.g., cell volume, Hog1, glycerol) and
using varied input waveforms (e.g., steps and ramps), we assess the network location of the
mechanism responsible for perfect adaptation in a minimally invasive manner. We conclude that
the system contains only one effective integrating mechanism, which requires Hog1 kinase
activity and regulates glycerol synthesis but not leakage.
INTRODUCTION
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Positive and negative feedback loops are ubiquitous regulatory features of biological
systems in which the system output reinforces or opposes the system input, respectively.
Quantitative models are increasingly being used to study the function and dynamic
properties of complicated, feedback-laden biological systems. These models can be broadly
classified by the extent to which they represent specific molecular details of the network. At
one extreme are the exhaustive models that dynamically track quantities of virtually all
biomolecules in a system, often using differential equations based either on known or
assumed reaction stoichiometries and rates. At the other end of the modeling spectrum is the
minimalist approach, which aims to fit and predict a system’s input-output dynamics with
only a few key parameters, each potentially the distillation of a large group of reactions.
*
Corresponding author: avano@mit.edu; 617-253-4446.
4These authors contributed equally to this work
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Examples of exhaustive and minimalist modeling approaches illustrate their unique
advantages and disadvantages. For instance, an exhaustive model of EGF-receptor
regulation (Schoeberl et al., 2002) can impressively predict the dynamics of 94 specific
network elements, but it requires nearly 100 parameters—some of which are not easily
measured biochemically—and could suffer from the omission of important reactions not yet
biologically identified. By contrast, minimalist models frequently lack such potentially
desirable reaction- and network-specific details, yet they excel at providing intuitive and
general insights into the dynamic properties of recurrent system architectures. For instance,
two elegant studies of bacterial chemotaxis—a system said to “perfectly adapt” because
abrupt changes in the amount of ligand only transiently affect the tumbling frequency,
whereas steady-state tumbling is notably independent of the ligand concentration—
highlighted a general feature of all perfectly adapting systems (Barkai and Leibler, 1997; Yi
et al., 2000). Specifically, it was shown that a negative feedback loop implementing
“integral feedback” is both necessary and sufficient for robust perfect adaptation in any
biological system (Yi et al., 2000). Mathematically, a dynamic variable (e.g., x or [cyclinB]) is an “integrator” if its rate of change is independent of the variable itself (e.g., if the dx/
dt and d[cyclin-B]/dt equations contain no terms involving x or [cyclin-B], respectively),
and integral feedback describes a negative-feedback loop that contains at least one
integrator. Biologically, a biomolecule acts as an integrator if its rate equation is not a
function of the biomolecule concentration itself; such a situation arises if, say, the synthesis
and degradation reactions are saturated (Supplemental Data). By providing specific
mechanistic constraints that apply to any perfectly adapting system, these two studies
underscored the function and significance of perfect adaptation in homeostatic regulation
and demonstrated the power of the minimalist modeling approach.
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Both exhaustive and minimalist modeling tactics have been successfully applied to the
osmosensing network in the budding yeast Saccharomyces cerevisiae (Klipp et al., 2005;
Mettetal et al., 2008). The core of this network is a highly conserved mitogen-activated
protein kinase (MAPK) cascade, one of several such cascades in yeast that regulate
processes ranging from mating to invasive growth while being remarkably robust to crosstalk despite their many shared components (Hohmann, 2002; Schwartz and Madhani, 2004).
Yeast cells maintain an intracellular osmolarity in excess of the extracellular osmolarity,
thereby creating positive turgor pressure across the cell wall and membrane that is required
for many processes including budding itself. Sudden drops in turgor pressure, potentially
caused by an upward spike in the external osmolyte concentration, are detected by
membrane proteins such as Sln1 (Reiser et al., 2003), which rapidly initiates a MAPK
cascade culminating in the dual phosphorylation of the MAPK Hog1 (Figure 1A). Upon
dual phosphorylation, the normally cytoplasmic and inactive Hog1 becomes activated and
translocates to the nucleus (Ferrigno et al., 1998), where it plays direct and indirect roles in a
broad transcriptional response (O’Rourke and Herskowitz, 2004). Glycerol-producing
factors are among the activated genes, and they facilitate osmoadaptation through the
increase of intracellular osmolarity (Hohmann et al., 2007). In fact, glycerol accumulation
has been shown to comprise 95% of the internal osmolarity recovery (Reed et al., 1987).
The subsequent restoration of turgor pressure leads to nuclear export of Hog1, which is
dephosphorylated by several nuclear and cytoplasmic phosphatases.
Non-transcriptional mechanisms also play an important role in the hyperosmotic-shock
response. Some are independent of Hog1 (e.g., rapid closure of Fps1 channels, which
otherwise allow passive leakage of glycerol (Luyten et al., 1995; Tamás et al., 2000)), but
others involve feedback mediated by the Hog1 pathway (Dihazi et al., 2004; Proft and
Struhl, 2004; Westfall et al., 2008). For instance, in response to hyperosmotic stress, the
glycolytic protein Pfk26, which stimulates production of glycerol precursors, was found to
be activated via phosphorylation at MAPK consensus sites in a Hog1-dependent manner
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(Dihazi et al., 2004). Additionally, in a recent study by Thorner and colleagues, it was
shown that Hog1 sequestered in the cytoplasm can still mount an effective osmotic response
(Westfall et al., 2008).
Biochemical characterization of most systems is rarely so rich to be deemed exhaustive, nor
so minimal to consider a system as a black box. Thus, models combining elements from
both approaches can be quite useful, such as a recent study that started with the exhaustive
osmoadaptation model (Klipp et al., 2005) and abstracted several elements to yield a
reduced representation (Gennemark et al., 2006). Here we take the reverse strategy, starting
instead with the minimalist model (Figure 1B) and then using biological measurements and
engineering principles to better understand systems-level dynamics and their relation with
network topology.
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In this study, we monitor the single-cell dynamics at high temporal resolution of both cell
volume and Hog1 nuclear enrichment simultaneously in response to hyperosmotic stress.
We observe perfect adaptation of Hog1 nuclear enrichment in response to step inputs of
osmolyte; this adaptation occurs with very low cell-to-cell variability and is robust to the
signaling fidelity of the MAPK cascade. From extensive theory developed in control
engineering, we know that perfect adaptation in this feedback system requires an integralfeedback mechanism. We refine the position(s) of integrator(s) in our network by generating
a range of putative network configurations and systematically rejecting those inconsistent
with our data. Facilitating this process of elimination is our observation that perfect
adaptation requires Hog1 kinase activity but not new protein production, suggesting that
Hog1 may implement integral-feedback via a yet-unknown role in protein-protein
interactions that increase the internal osmolyte concentration. Measurements of glycerol
accumulation suggest that this crucial role for Hog1 kinase activity upregulates glycerol
synthesis but does not otherwise regulate its leakage. Finally, in an experiment imposing
severe constraints on the possible valid network configuration, we show that neither cell
volume nor Hog1 nuclear enrichment perfectly adapts in response to a ramp input of salt.
Together, our results establish that the system’s negative feedback loop contains exactly one
effective integrating mechanism. Though the loop may branch such that this effective
integrator is composed of multiple integrating reactions arranged in parallel, we can reject
the possibility that the net feedback loop contains two or more effective integrating
mechanisms arranged in series.
RESULTS
MAPK Cascade Introduces Negligible Noise While Transducing Osmotic-Shock Signal
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To gain further insight into Hog1 signaling dynamics and the feedback mechanisms that
underlie osmoadaptation, we assayed the nuclear accumulation of Hog1 and cellular volume
in single cells over time in response to hyperosmotic shock. We fused a yellow-fluorescent
protein (YFP) to the C-terminus of endogenous Hog1 (Hog1-YFP) in haploid cells from
which SHO1 was deleted. SHO1 deletion disables one of the two primary branches that
activate Hog1, leaving the Sln1 branch as the main activator of Hog1. Importantly, SHO1
deletion eliminates crosstalk with other MAPK cascades (McClean et al., 2007; Schwartz
and Madhani, 2004) while still preserving the Hog1 dynamics (compare Figure 2A and
Figure S1, (Hersen et al., 2008)) and gene expression profiles (O’Rourke and Herskowitz,
2004) of cells where both branches are intact. To identify the nucleus and thereby facilitate
our computation of nuclear Hog1 enrichment, we also fused a red fluorescent protein to
Nrd1 (Nrd1-RFP), a strictly nuclear factor. Throughout this work, we refer to this strain as
“wildtype”.
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We created a simple flow-chamber apparatus to permit the changing of media with
simultaneous acquisition of images. Cells were grown to log phase and loaded into the
chamber (Figure 1C), where they adhered to a coverslip coated with concanavalin A. Media
with or without excess osmolyte was washed over the cells throughout each experiment, and
in less than two seconds the media within the chamber could be switched completely
between the two types (data not shown). At each time point, we acquired phase-contrast,
YFP, and RFP images, allowing us to observe a transient enrichment of Hog1 accumulation
in the nucleus shortly after a hyperosmotic shock and diminished localization shortly before
and long after the shock (Figure 1D).
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We developed custom image-analysis algorithms to quantify the dynamic decrease in singlecell volume and increase in Hog1 nuclear enrichment in response to a step increase (Figure
2A) of the extracellular osmolyte concentration. For each cell, we compute the ratio of
nuclear YFP signal to whole-cell YFP signal throughout the experiment, and define a cell’s
Hog1 nuclear enrichment as the relative change from the pre-shock level of this ratio
(Experimental Procedures). Single-cell volume dynamics were computed from
measurements of cell area (Experimental Procedures) and were interpreted as a proxy for
turgor pressure. Single-cell Hog1 and volume data from a representative flow-cell
experiment are posted on the Cell website. We observed remarkably low cell-to-cell
variability in the network response, which includes the reactions in the MAPK cascade: for
different cells, the amplitude and timing of changes in Hog1 nuclear enrichment and cell
volume are very similar, with trends that closely follow the population average (Figure 2A,
B). In fact, fluctuations in unstimulated cells were of a similar magnitude (Figure S2),
further indicating that the intrinsic noise of signal propagation is low and suggesting that the
experimental setup itself may be the predominant source of noise in our data. Compared
with the recent observation of significant cell-to-cell variability in the dynamic nuclear
enrichment of the yeast calcium-stress regulator Crz1 (Cai et al., 2008), these data suggest
that the osmoadaptation signaling system generates output signals with very low noise,
despite the fact that the system itself contains many proteins expressed at noisy levels
(Newman et al., 2006).
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Since our flow-cell-based assay allows for simultaneous measurement of cell volume and
Hog1 nuclear localization at unprecedented timescales, we sampled the osmotic response
every two seconds in response to a step of salt (Figure 2C, D). Again, we argue that the
MAPK cascade generates negligible noise since the cell-to-cell variability in both the input
(volume) and output (nuclear Hog1) are comparable. Together with the data from Figure 2A,
B, this data gathered every two seconds reveals four distinct regimes in the relationship
between cell volume and Hog1 activation in response to osmotic stress (Figure 2E). Within
the first 20 seconds following a step of salt, cell volume drops precipitously, but Hog1
nuclear enrichment remains at its basal level while the stress signal propagates through the
MAPK cascade. Between 20 seconds and 90 seconds, Hog1 nuclear enrichment rises from
its minimum to its maximum but does not yet affect turgor pressure, as cell volume remains
at its minimum. Shortly thereafter, turgor pressure begins to restore, and Hog1 nuclear
enrichment begins to fall, with both reaching their pre-shock levels simultaneously at 25
minutes post-shock. Finally, cell growth resumes, as the cell volume grows and Hog1
returns to its basal level.
Hog1 Nuclear Enrichment Perfectly Adapts
We found that the steady-state Hog1 nuclear enrichment is identical to its pre-stimulus level
over a range of hyperosmotic-shock strengths (Figure 3A). This dynamic behavior is the
hallmark of perfect adaptation. Importantly, Hog1 perfect adaptation is not NaCl-specific, as
we observed it with KCl and sorbitol treatment as well (Figure S3). In our simultaneous
measurements of cell volume and Hog1 localization, we found a very strong correspondence
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between the timing of both Hog1 adaptation and the restoration of turgor pressure, which we
interpret as the moment that the rate of volume growth (i.e., slope of volume curve) in
stressed cells matches that of unstressed cells (Figure 3A, B). Interestingly, for cells stressed
with media containing 0.6M NaCl, this turgor pressure recovery appears to occur at a
volume less than the pre-shock cell volume. This observation is consistent with membrane
invagination that has been shown previously to occur only in cells under acute hyperosmotic
stress (Slaninová et al., 2000): by decreasing membrane surface area, a particular turgor
pressure can be achieved at a smaller volume.
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We conclude from these data that the Hog1 nuclear enrichment level perfectly adapts and
that this behavior is tightly coupled with perfect adaptation of turgor pressure. Taken
together with extensive theoretical analysis of adaptive systems in engineering disciplines,
these findings require that the system implement integral-feedback control (Ingalls et al.,
2006; Sontag, 2003; Yi et al., 2000). The presence of integral feedback makes perfect
adaptation a robust system feature that does not require a careful tuning of the system
parameters, such as protein levels or rate constants. To demonstrate the robustness of this
perfect adaptation, we measured the Hog1 response after changing the signaling fidelity of
the MAPK cascade by controlling the expression of PBS2, which encodes the kinase of
Hog1. To this end, we placed the genomic copy of PBS2 under the inducible control of a Tet
promoter. At saturating doxycycline levels, we observed Hog1-nuclear-enrichment
dynamics comparable with wildtype cells (data not shown). By contrast, at low-level
induction of the Tet promoter, the amplitude of the Hog1 response was significantly less
than in wildtype, and we did not observe saturation in the peak amplitude as a function of
salt (Figure 3C). Despite these gross differences, however, Hog1 nuclear enrichment still
perfectly adapted. We conclude that integral feedback is a structural feature of the network;
thus, perfect adaptation is a robust property of the system and not a consequence of precisely
tuned parameters.
Strategy for Finding the Integrator
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Our data begin to restrict the possibilities of where in the network the integrator can be. The
potential locations we consider are four subsystems denoted H, I, D, and G (Figure 4). The
H subsystem represents all relevant reactions that link an osmotic disturbance at the
membrane with Hog1 nuclear enrichment; for example, the MAPK cascade and nuclearimport reactions are in this subsystem. In the D subsystem are Hog1-dependent mechanisms
that promote glycerol accumulation, such as the transcriptional activation of genes that
encode glycerol-producing enzymes and potential protein-protein interactions initiated by
Hog1 in the cytoplasm or nucleus that lead to glycerol accumulation. In subsystem I are the
Hog1-independent mechanisms that contribute to osmolyte production, such as exportchannel closure and gene-expression mediated by stress factors other than Hog1. Finally,
subsystem G represents the metabolic reactions involved in glycerol synthesis and any other
reactions that contribute to glycerol accumulation. Positing that each subsystem either
contains one or more integrators or contains none, we generated the 16 possible network
configurations to guide our further analysis (Figure 4).
A critical aid in finding which of the subsystems contains the integrator(s) is the fact that
with respect to the furthest-downstream integrator in a feedback loop, quantities upstream
perfectly adapt, and those downstream do not (Supplemental Data). For instance, the
observation that turgor pressure perfectly adapts—indeed, this is arguably the primary
function of regulating osmotic stress—stipulates that at least one integrator must exist in H,
I, D, or G, since turgor pressure is upstream of each of these systems within the feedback
loop. This allows us to reject network configuration (a), where none of the subsystems acts
as an integrator. The observation of Hog1 perfect adaptation imposes an additional
constraint: if the only integrator in the feedback loop were in subsystem H, then Hog1
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nuclear enrichment would not perfectly adapt, because it is downstream of H. Thus, at least
one of the subsystems I, D, and G must contain an integrator, permitting the rejection of
configuration (b).
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The high-temporal resolution of our Hog1 measurements provides another severe constraint
on possible network configurations. Our assay permits precise quantification of the area
under the Hog1 curve (i.e., its integral). We find that in both wildtype cells and PTet-PBS2
mutant cells, integrated-Hog1 scales linearly with the shock strength (Figure 3D) (note: the
fact that their slopes differ does not affect the subsequent argument). If the system were
composed only of reactions that could be modeled with linear dynamics, then such a result
would be trivial; however, the fact that the peak Hog1 amplitude saturates as a function of
salt (Figure 3A) is strong evidence of nonlinearity in the H subsystem and makes the data in
Figure 3D quite informative. In particular, these data suggest that exactly one integrator
exists between the input to subsystem D and the output of subsystem G, in addition, perhaps,
to other serially arranged non-integrating subsystems with nearly linear input-output steadystate characteristics (Supplemental Data). We can reject the scenario in which both D and G
are integrators since the double integral of Hog1 would be expected to scale linearly with
shock strength, which is clearly not the case (Figure 3D, inset). Conversely, if neither D nor
G is an integrator, then the fact that Hog1 perfectly adapts means that I must be an
integrator. Not only do we experimentally refute this possibility in the next section, but we
can also theoretically dismiss it because the nonlinearity in H would eliminate the linear
scaling we observe between shock strength and integrated-Hog1. Thus, we conclude that
either D is an integrator or G is an integrator, allowing us to reject configurations (a), (b),
(c), (f), (k), (n), (o), and (p).
Perfect Adaptation Requires Hog1 Kinase Activity, Which Affects Glycerol Synthesis but
not Leakage
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To distinguish among the remaining putative network configurations, it was important to
separate the roles of Hog1-dependent (D subsystem) and Hog1-independent (I subsystem)
reactions. To this end, we mutated the endogenous Hog1 gene in our wildtype strain such
that the kinase activity of the mutant (hog1-as) could be specifically, rapidly, and inducibly
ablated by the ATP-analog 1-NM-PP1 (“PP1”) (Westfall and Thorner, 2006). PP1 treatment
has been shown to completely eliminate Hog1 kinase activity, unlike many of the previously
studied constitutively “dead” isoforms, which were shown to retain both residual kinase
activity (Westfall and Thorner, 2006) and the ability to perfectly adapt (Figure S4). In the
presence of PP1, perfect adaptation is lost, as steady-state Hog1 accumulation does not
return to its pre-stimulus level (Figure 5A). These results and those in Figure 5B were
corrected for the effect PP1 has on cells not shocked with salt to isolate the salt-specific
response (Figure S5, Experimental Procedures). Though the peak amplitude is less than in
cells untreated with PP1 (Figure 5A, inset), this failure of PP1-treatred cells to adapt is not
an artifact of the documented defect in nuclear transport of hog1-as (Westfall and Thorner,
2006) because nuclear enrichment dropped precipitously upon removal of the stimulus
(Figure S6). Furthermore, the effect is specific to Hog1-kinase activity, since cells
expressing wildtype Hog1 perfectly adapted even in the presence of PP1, though they, too,
had diminished peak amplitude (Figure S7). Importantly, PP1-treatment eliminates the postshock volume recovery (Figure 5B), suggesting that kinase-dead Hog1 persists in the
nucleus because turgor pressure is not restored. We conclude that Hog1 kinase activity is
necessary for the proper functioning of the integral-feedback controller in the
osmoadaptation network.
Since PP1-treatment effectively severs the feedback loop between nuclear Hog1 and the
Hog1-dependent mechanisms (i.e., it disconnects the D subsystem from its input), the loss of
perfect adaptation further constrains the possible locations for the integrator(s). First, the
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Hog1-independent subsystem (the I subsystem) cannot contain the last integrator in the
feedback loop. If it did, then the turgor pressure would perfectly adapt in the presence of
PP1, and Hog1 likely would as well, yet we observe neither (Figure 5A, B); thus, we reject
scenarios (c), (f), (i), and (l) from Figure 4. We can similarly reject scenarios where both I
and G (the glycerol-accumulation subsystem) act as integrators (i.e., scenarios (j), (m), (o),
and (p), Figure 4), since these scenarios would also ensure perfect adaptation in the system
in the presence of PP1, which was not observed.
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We expected that the failure of Hog1 nuclear enrichment to perfectly adapt in the absence of
Hog1 kinase activity resulted from insufficient glycerol accumulation. Therefore, we
compared internal glycerol accumulation in cells either treated or not treated with PP1. In
cells not treated with PP1 (“−PP1”), intracellular glycerol rose rapidly after hyperosmotic
shock but remained constant in unshocked cells (Figure 5C). In PP1-treated cells (“+PP1”),
hyperosmotic shock caused a slight increase in internal glycerol, but the level achieved was
significantly less than in −PP1 cells, consistent with loss of both turgor-pressure restoration
and Hog1 perfect adaptation. There was a qualitative similarity between the traces of Hog1
nuclear enrichment and the rate of internal glycerol accumulation in −PP1 cells (compare
Figure 3A and Figure 5D). Like the linear scaling of integrated-Hog1 with salt shock (Figure
3D), this correspondence is expected from network scenarios where exactly one integrator
(in addition, perhaps, to some appropriate upstream and/or downstream linear subsystems)
exists between Hog1 and glycerol. In the simplest case where the subsystem between Hog1
and glycerol contains only a single integrator and nothing else, the rate of glycerol
accumulation would effectively be the derivative of the integral of Hog1 nuclear enrichment
and should, therefore, be simply a scaled version of the Hog1 curve as observed. In +PP1
cells, where perfect adaptation is lost, this correspondence is also lost (compare Figure 5A,
D) as the Hog1 and glycerol-accumulation-rate curves diverge at their post-stimulus steadystate levels (i.e., the rate of glycerol accumulation drops to zero, whereas Hog1 nuclear
enrichment remains nonzero).
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Insufficient glycerol accumulation in +PP1 cells prompted us to investigate the ways in
which glycerol synthesis and leakage depend on Hog1 kinase activity. We measured celldensity-normalized levels of total glycerol and extracellular glycerol over time in the
presence and absence of osmotic shock and PP1 (Figure 5E, F). In −PP1 cells, hyperosmotic
shock leads rapidly to a transient decrease in glycerol leakage (Figure 5E) and a persistent
increase in glycerol synthesis (Figure 5F) as compared to cells unstressed with salt. The
results in +PP1 cells are partially influenced by a nonspecific effect of PP1. In the absence
of salt shock, +PP1 cells increase glycerol synthesis and equivalently increase glycerol
leakage, a behavior we found to be independent of Hog1 since it also occurred in cells with
wildtype Hog1 (data not shown). Nonetheless, in +PP1 cells treated with salt, glycerol
leakage is rapidly and transiently diminished, just as in −PP1 cells; yet the absence of Hog1
kinase activity prevents an increase in glycerol synthesis, unlike in −PP1 cells. Together,
these data suggest that Hog1 kinase activity plays a critical role in rapidly regulating
glycerol synthesis but not its leakage, consistent with a recent report (Westfall et al., 2008).
Neither Cell Volume Nor Hog1 Perfectly Adapts in Response to a Ramp Input
Four putative network configurations remain (Figure 4, (d), (e), (g), and (h)), and we use an
experiment motivated by control engineering to restrict the options even further. The key
concept is that the number of integrators arranged in series in a system can be deduced by
assaying for perfect adaptation in response to very specific input trajectories. In general,
perfect adaptation to an input corresponding to the nth-integral of a step function implies
that the system contains at least one feedback loop with at least n+1 integrators arranged in
series, where n is a positive integer. For the step input itself, where n=0, perfect adaptation
demonstrates the existence of one or more integrators arranged in series. Perfect adaptation
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in response to a linear-ramp input (n=1, since a ramp is simply the integral of a step) reveals
the existence of two or more integrators in series, and so on.
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We performed a flow-cell experiment in which the salt concentration ramps upward over
time (Figure 6A), reaching a plateau after nearly 45 minutes. The slope of our ramp was
chosen to reconcile two opposing factors: (1) that the ramp be steep enough to observe a
signal in both volume and Hog1, and (2) that the ramp be shallow enough that both volume
and Hog1 can reach steady state before the exterior salt concentration becomes so toxic that
it occludes the response specific to the hyperosmotic shock system. In response to our ramp
input and critically before the beginning of the plateau, we observed that neither volume nor
Hog1 perfectly adapted (Figure 6B, C). Importantly, shortly after the plateau begins, both
volume and Hog1 return to pre-stimulus levels, indicating that their failure to adapt in the
presence of the ramp is specific to the stimulus itself and not because of the high final salt
concentration of 0.75M.
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The cell volume clearly reaches steady state within the ramp period and does not adapt, but
it could be argued that Hog1 nuclear enrichment does not achieve steady state and could
actually perfectly adapt were the ramp to persist for longer. Since even a pessimistic
extrapolation of the pre-plateau Hog1 data would lead to perfect adaptation only after more
than two hours, we argue that a second integrator permitting such adaptation is largely
irrelevant because the extracellular salt concentration would be in excess of 2M at the time
of adaptation, and the cells would almost certainly have perished.
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In combination with our other findings, our ramp-input results confirm that there is exactly
one integrator arranged in series in the system. This finding rejects many model
configurations (Figure 4), but critically it invalidates (g) and (h), which had not been
otherwise refuted. We argue that the most likely network configuration is (d), where
subsystem D functions as an integrator and G does not. If G were to act as an integrator,
then cell volume and turgor pressure would continue to recover as long as the input to the G
subsystem is nonzero. But, when subsystem D is disabled in the presence of PP1, the only
input to subsystem G is the output from subsystem I. Thus, the observed failure in volume
recovery in PP1-treated cells (Figure 5B) would only occur if the output of subsystem I
prematurely goes to zero (i.e., if it were a “differentiator”). This observation would require
that all Hog1-independent mechanisms completely desensitize within approximately 20
minutes (i.e., the time needed for hog1-as nuclear enrichment to reach steady state; Figure
5B) despite persistence in their stimulus (i.e., the acute loss of turgor pressure). On the basis
of this argument, we consider it extremely improbable that subsystem G acts an integrator.
Therefore, we reject scenario (e), leaving (d) as the last remaining configuration. We used a
system of coupled differential equations to implement this concise model and found that it
can indeed recapitulate the key features of our data (Figure S8, Supplemental Data). In sum,
we conclude that the integrator responsible for perfect adaptation in response to a step input
is a Hog1-dependent mechanism upstream of glycerol synthesis.
DISCUSSION
Using both biochemical techniques and engineering principles, we demonstrate that the lownoise (Figure 2), robust perfect adaptation of Hog1 nuclear enrichment and cell volume
(Figure 3) results from a single integrating mechanism (Figure 4, 6) that requires Hog1
kinase activity (Figure 5A, B) and regulates glycerol synthesis (Figure 5E, F). Interestingly,
despite a well-known role in mediating an osmostress-induced transcriptional response,
Hog1 does not require gene expression to perfectly adapt (Figure S9, (Westfall et al., 2008)).
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It is worth noting that our results are not inconsistent with multiple Hog1-kinase-dependent
integrating reactions operating in parallel (our analysis here specifically rejects the
possibility of multiple integrators in series). Redundant parallel integrating reactions would
nonetheless yield one “effective” integrating mechanism, where each individual integrating
reaction would strengthen the effective integrator and make the adaptation more rapid.
Compared to a single integrating reaction, a parallel arrangement would also make the
system more robust to mutation, since disabling one of the parallel integrating reactions
would simply delay adaptation rather than prevent it altogether. Although future
experiments will be required to determine exactly which reaction(s) is an integrator, the
search may be facilitated by considering reactions downstream of Hog1 that are most likely
to operate at saturation, since saturated reactions are one way to biologically implement
integration (Supplemental Data) (Gomez-Uribe et al., 2007). We tested whether the series of
reactions involving Hog1, Pfk26, and Pfk1,2 could be the integrator. Pfk26, which is
activated in a Hog1-dependent manner after hyperosmotic stress (Dihazi et al., 2004),
synthesizes factors that enhance the production of glycerol precursors by the highly
abundant—and potentially saturating—Pfk1,2 complex. We knocked out PFK2, which has
been shown to decrease the Pfk1,2 complex activity by nearly 95 percent (Arvanitidis and
Heinisch, 1994) but saw no loss of perfect adaptation or appreciable delay in the adaptation
kinetics (Figure S10), suggesting that this chain of reactions does not implement integral
feedback. Other highly expressed substrates of Hog1 (Kim and Shah, 2007) should be
investigated in the future for integrating properties.
A Dynamic Perspective on Hyperosmotic Shock Recovery
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Our results yield the following dynamic portrayal of osmoregulation (Figure 7). In the
absence of hyperosmotic shock, cells maintain a constant internal glycerol concentration.
Glycerol is synthesized at a rate slightly higher than that required to keep pace with cell
growth, and some glycerol is leaked (Figure 7, panel i). Within 20 seconds after a
hyperosmotic shock, the cell volume drops to its minimum (Figure 2D), but Hog1 is not yet
beginning to be enriched in the nucleus (Figure 2C), though some cytoplasmic Hog1 likely
becomes activated and begins to upregulate glycerol synthesis (Figure 7, panel ii). After 20
seconds, the volume remains at its minimum but Hog1 becomes maximally enriched in the
nucleus and glycerol synthesis continues to increase (Figure 7, panel iii). Beyond the first 5
minutes (Figure 7, panel iv), cell volume rises (Figure 2B and 3B), nuclear Hog1
accumulation begins to fall (Figure 2A and 3A), glycerol continues to be synthesized
(Figure 5F), and some glycerol begins to leak to the exterior (Figure 5E). It is noteworthy
that internal glycerol continues to increase beyond 15 minutes in −PP1 cells (Figure 6A) but
not in +PP1 cells, indicating that +PP1 cells prematurely leak as much glycerol as they
synthesize. If leakage prevention were stronger and/or endured for longer, then these cells
could also accumulate enough glycerol to perfectly adapt; these data suggest that Hog1
kinase activity may play a role in the maintenance of glycerol retention beyond 15 minutes,
though its effect may be indirect.
Beyond 25 minutes (Figure 7, panel v), the Hog1 nuclear enrichment returns to its prestimulus level (Figure 2A and 3A), the hallmark of perfect adaptation. Although the pre- and
post-stimulus rates of glycerol accumulation are identical (Figure 6C, D), glycerol synthesis
and leakage are significantly higher after the stimulus than before it (Figure 6E, F).
A Systematic Approach for Biological Systems Analysis
Our modeling efforts here and in Mettetal et al. underscore the power of applying
engineering principles to any biological system that has well-defined and quantifiable
inputs, internal variables, and outputs. In analyzing such systems, measurement of the
frequency response (Mettetal et al., 2008) provides an estimate of the number of relevant
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dynamic variables that a minimal model should have; additionally, these measurements may
reveal the basic computation or function that the system performs. For systems potentially
involving hundreds of reactions, each with unique kinetics, such analysis can appreciably
reduce the complexity of a model without compromising, and potentially enhancing, the
insight into systems-level behavior. Once the minimum number of dynamic variables has
been estimated, the existing biochemical knowledge of the system can be leveraged to infer
which biological quantities correspond to the relevant dynamic variables, and to create a
basic network diagram such as the one in Figure 1B. Candidate molecules that may
correspond to the relevant dynamic variables should include (1) factors known to be
important in the system (e.g., glycerol and cell volume for the yeast osmotic-shock
response), and (2) factors whose dynamics are expected to change on an appropriate
timescale (e.g., fast protein-protein interactions versus slow gene-expression events). By
measuring the dynamics of the internal variables in the system’s network diagram, the
model can be further constrained, and links can be drawn more confidently between model
elements and biological mechanisms.
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Perfect adaptation is one such dynamic feature that restricts potential models, where a basic
result from control engineering provides important information about biological mechanism
(Stelling et al., 2004). We expect similar analyses to be particularly useful in the study of
other homeostatic systems that are ubiquitous in biology (e.g., blood calcium levels (ElSamad et al., 2002)), where perfect adaptation is presumably the paramount dynamic
property of the network. Most of these systems likely achieve robust perfect adaptation
through a negative feedback loop with one or more integrators; thus, an important future
endeavor is to better understand how biological systems implement integration at the
molecular level. Since the simple loss of integral feedback can fundamentally transform the
function of a system—from one with transient output into one with persistent and potentially
deleterious output—identifying and characterizing biological mechanisms providing integral
feedback should be instrumental in the future study of homeostatic systems, the design of
perfectly adapting biosynthetic circuits, and the development of therapeutics to combat
disease.
EXPERIMENTAL PROCEDURES
Strain Background and Construction
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Our haploid “wildtype” strain (DMY017) was derived from the DMY007 strain (Mettetal et
al., 2008), the only difference being that the SHO1 ORF was excised via standard PCRbased methods. Our PP1-sensitive strain (DMY034) was created by deleting the HOG1 ORF
from DMY017 (DMY030), generating a single-base mutation of HOG1 on a plasmid using
QuikChange (Stratagene), and then inserting the resulting hog1-as ORF (Westfall and
Thorner, 2006) into DMY030 using PCR integration.
Fluorescence Microscopy and Image Analysis
Flow-chamber construction, cell preparation, image acquisition and image segmentation
were all performed as described (Mettetal et al., 2008). For ramp experiments, a syringe
pump (Harvard Apparatus) dispensed media containing concentrated NaCl into a graduated
cylinder containing fresh media agitated by a magnetic stir bar. A peristaltic pump pulled
liquid directly from the graduated cylinder into the flow cell. In both step and ramp
experiments, each cell in an experiment is assigned an index. To quantify the Hog1 response
of cell number i we compute the cell’s raw nuclear enrichment ri(t), defined as the ratio of
nuclear YFP intensity to whole-cell YFP intensity (a comparison of different quantification
methods is in Supplemental Data, Figure S11). Specifically, the raw nuclear enrichment is
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given by
, where the numerator is the average of the ten brightest nuclear
pixels—unlike the average of the whole nucleus, this top-ten metric compensates for the
diminishing ability to properly segment nuclear boundaries as Nrd1-RFP bleaches—and the
denominator is the average intensity within the entirety of cell i.. Defining ri(t) as a ratio of
YFP intensities provides some correction for photo-bleaching.
We define the Hog1 nuclear enrichment, hi(t), of cell number i by
, where
ri(t0) is the raw nuclear enrichment of cell number i before osmostress. We estimate ri(t0) by
the average raw nuclear enrichment for the four preshock time points.
To quantify single-cell volume, we first compute the number of pixels ni(t) that constitute
cell number i as a function of time. This number is proportional to the cell’s area, so
assuming that the cell is a sphere, (ni(t))1.5 is directly proportional to the volume. We define
the relative volume increase, vi(t), of cell number i by
. Here ni(t0) is the
pre-shock number of pixels of cell number i, which we estimate by taking the average of
ni(t) over the four pre-shock time points.
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To correct the Hog1 and volume data from our +PP1 and +CHX experiments, we subtract
the mean trace (N≥3) in unstressed cells (i.e., an hour-long experiment with only 0M
[NaCl]; Figure S5) from the mean trace in stressed cells (N≥3). We assume the error in
stressed and unstressed experiments is independent, so we add the respective variances and
then calculate the standard error of our corrected mean trace.
Glycerol Assays
Glycerol levels were measured using the Free Glycerol Reagent Kit (Sigma). For details
about preparing cells and separately acquiring intracellular, extracellular, and total glycerol
levels, see Supplemental Data.
Supplementary Material
Refer to Web version on PubMed Central for supplementary material.
Acknowledgments
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We thank J. Falvo, R. Tsien, and E. O’Shea for the mRFP1.3 plasmid, as well as Suzanne Komili, George Verghese
and Leonid Mirny for insightful comments and discussions. This work was supported by NSF Graduate Research
Fellowships to D.M. and J.T.M., a MIT-Merck Graduate Fellowship to C.A.G-U., and NIH grants R01-GM068957
and 5 R90 DK071511-01.
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Figure 1. Hog1 translocates to the nucleus in response to hyperosmotic shock
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(A) An upward spike in external osmotic pressure at t=0 is sensed by Sln1, which
propagates a signal via a MAPK cascade that culminates in the dual phosphorylation and
nuclear import of Hog1. In the nucleus, Hog1 activates gene expression; active Hog1 also
plays various roles in the cytoplasm to facilitate osmoadaptation. Together with Hog1independent stress responses, these mechanisms increase internal glycerol, thereby restoring
turgor pressure across the membrane. Our “wildtype” strain has YFP fused to Hog1, and the
Sln1 branch is the primary osmosensor, since SHO1 is deleted.
(B) Network diagram inferred from (A) showing the system input and measurable outputs.
(C) Schematic of our simple flow chamber. The pump draws media through the flow-cell,
with the osmolarity controlled by a valve. Cells adhere to the coverslip coated with 1 mg/ml
conA in water.
(D) Phase-contrast (top row), Hog1-YFP (middle row), and Nrd1-RFP (bottom row) images
are captured at each time point. In the absence of hyperosmotic shock, Hog1-YFP is
distributed throughout the cell (left column). Only transiently does Hog1 accumulate
appreciably in the nucleus after a shock (middle column), restoring the pre-stress
distribution at steady state (right column).
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Figure 2. MAPK signaling introduces very little cell-to-cell variability in Hog1 nuclear
enrichment
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(A) In response to a step shock of 0.4M NaCl, dynamics in single-cell Hog1 nuclear
enrichment (shown for three cells in green, orange, and purple; defined in Experimental
Procedures) strongly resemble the population average (dark-blue dotted line) obtained from
a single experiment (>100 cells). The shaded region is +/− STD around the population
average and represents the cell-to-cell variability of a single experiment.
(B) Cell volume traces are low in noise (salt stimulus, single cells, mean, and shaded area
are as in (A)). After an abrupt decrease in cell volume after t = 0, volume recovers and
exceeds the pre-stimulus value, indicative of cell growth within the flow cell once turgor
pressure is restored.
(C) and (D) Two-second sampling of single-cell Hog1 nuclear enrichment (C) and cellular
volume (D) (salt stimulus, single cells, mean, and blue shaded area are as in (A)). Yellow
shaded region depicts +/− SEM for >3 independent flow cells.
(E) For each time point in panels (A) through (D), the cell volume and Hog1 nuclear
enrichment values are positioned on this scatter plot of Hog1 versus volume, illustrating four
distinct regions. Spots are false-colored for clarity, becoming black near the boundaries
between regions.
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Figure 3. Hog1 nuclear enrichment exhibits robust perfect adaptation
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(A) In response to hyperosmotic shocks with indicated concentrations of NaCl, the steadystate Hog1 nuclear enrichment returns exactly to the same level as in unshocked cells. All
cells were grown and loaded into the flow cell using media with 0.0 M NaCl. Dotted lines
show the average response, obtained by averaging population averages from independent
experiments (N=3 or 4); error boundaries depict +/− SEM.
(B) Cell volume traces corresponding to the experiments in (A), except that all post-shock
data points are the average of measurements at three consecutive time points.
(C) Hog1 nuclear enrichment perfectly adapts even in cells with compromised MAPK
signaling. The TetO7 promoter was integrated upstream of the endogenous PBS2 gene. In
the experiments shown, [doxycycline] = 0.06 μg/ml.
(D) The area under the Hog1 curves in (A) and (C) scales linearly with [NaCl] consistent
with a single integrator between Hog1 activation and glycerol accumulation (Supplemental
Data); dotted gray lines are guides for the eye. Data points indicate mean +/− STD of the
area calculated from at least three independent Hog1-nuclear-enrichment traces. The inset
shows the double integral of Hog1 curves in (A), calculated up to the point where the data
curves adapt.
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Figure 4. Perfect adaptation and other data reject many potential network schematics
We permute the model from Figure 1B (shown in reduced form at top-left) such that each of
the subsystems H, D, I, and G, is an integrator (shaded orange) or not (shaded green),
yielding the sixteen possibilities (a) through (m). For the reasons listed at top-right, which
consider data from Figures 2, 3, 5, and 6, certain network schematics can be rejected. Only
network (d) satisfies all the constraints.
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Figure 5. Hog1 kinase activity is required for perfect adaptation of Hog1 nuclear enrichment
and glycerol accumulation via a role in upregulating glycerol synthesis
(A) Nuclear enrichment of (hog1-as)-YFP in cells treated with 24 μM PP1 prior to
hyperosmotic shock with 0.4 M NaCl. This trace is corrected to highlight the salt-specific
response (Figure S3). In the inset, the same strain is stressed in the same manner, except PP1
is omitted. In both the main curve and the inset, the mean (N=3) +/− SEM is plotted.
(B) Mean volume traces (N=3) +/− SEM from the corresponding experiments in (A), again
corrected to highlight the salt-specific response (Figure S3).
(C) Dynamics in the concentration of intracellular glycerol (OD540 measurement from
glycerol kit; see Experimental Procedures), corrected for cell-growth (OD600 measurement)
by taking the ratio, were measured in hog1-as cells in the presence and absence of 24 μM
PP1 (administered 30 minutes before experiment) and hyperosmotic shock with 0.4 M NaCl
(administered at t = 0). Data points represent the mean (N=3) +/− STD.
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(D) The glycerol accumulation rate was computed as the slope of the traces in (A), and the
time of each plotted point corresponds to the midpoint of the two time points used in the
slope computation (e.g., the slope between the 0-minute and 15-minute time points is plotted
at 7.5 minutes). Error bars represent the combined standard deviation from data points in
(A), assuming measurements were independent at different time points.
(E) and (F): Dynamics in the extracellular (E) and total (F) glycerol concentration (see
Experimental Procedures) in experiments outlined in (C) and described in the main text.
Data points and error bars are as described in (C).
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Figure 6. Neither cell volume nor Hog1 nuclear enrichment perfectly adapts in response to a
ramp input
(A) The calculated concentration of NaCl in the flow chamber as a function of time (see
Experimental Procedures). We measured the fluorescence intensity of rhodamine diluted in
water and converted to [NaCl] based on the respective dilutions. The mean (N=3) +/− SEM
is shown.
(B) and (C) Temporal changes in relative volume increase (B) and Hog1 nuclear enrichment
(C) in response to ramp input shown in (A) (mean and error are same as in (A)). During the
ramp period, volume and Hog1 fail to perfectly adapt, but they later adapt during the plateau
period.
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Figure 7. A simple model based on our data captures key data features
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Cartoon representation of key features of yeast hyperosmotic shock response, explained in
detail in the Discussion. Hyperosmotic shock is applied at t = 0. Green membrane channels
are open, allowing the transport of glycerol, shown as purple circles. Upon closure, the
channel is depicted in red.
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